Image Processing Reference
In-Depth Information
Example 3.6
Find the inverse Laplace transform of the following function of the right-sided
signal x(t)if
s
X(s)
¼
1)
2
(s þ
(s þ
2)
S
OLUTION
Using partial-fraction expansion, we would have
s
A
1
s þ
A
2
(s þ
B
1
s þ
X(s)
¼
2)
¼
1
þ
1)
2
þ
1)
2
(s þ
2
(s þ
where
s
(s þ
B
1
¼
lim
s!
2
(s þ
2)X(s)
¼
lim
s!
1)
2
¼
2
2
s
s þ
1)
2
X(s)
A
2
¼
lim
s!
(s þ
¼
lim
s!
2
¼
1
1
1
1)
2
X(s)
ds
d(s þ
d
ds
s
s þ
2
(s þ
A
1
¼
lim
s!
¼
lim
s!
2
¼
lim
s!
2)
2
¼
2
1
1
1
Therefore,
2
1
2
X(s)
¼
1
1)
2
s
þ
s
þ
2
(s
þ
Hence,
[2e
t
te
t
2e
2t
]u(t)
x(t)
¼
Example 3.7
Solve the following second-order DE using Laplace transform. The initial condi-
tions are y(0)
1 and
dy(t)
dt
¼
j
t¼0
¼
4.
d
2
y(t)
dt
2
3
dy(t)
¼ e
4t
u(t)
þ
dt
þ
2y(t)
where u(t) is the unit step function.
S
OLUTION
Taking Laplace transform from both sides of the above DE, we have
1
s þ
s
2
Y(s)
y
0
(0)
sy(0)
þ
3[sY(s)
y(0)]
þ
2Y(s)
¼
4
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